Inequalities for norms and numerical radii of operator matrices

In this paper, we aim to derive a range of numerical radius inequalities. Our work not only confirms recently demonstrated numerical radius inequalities, but also introduces more precise numerical radius inequalities compared to those previously established for specific cases. Additionally, we delve into additional properties of operator matrices and provide fresh estimates for the operator norms and numerical radii of these operators. Moreover, we establish various upper bounds for the numerical radii of \(2\times 2\) $2\times 2$ operator matrices, refining and expanding the bounds previously determined.